I think I have managed to get the ACA620 99% stable now. I did a third, and extra thorough, cleaning of the CPU pins, and now I am able to twist the motherboard with the ACA620 installed, without crashes. Pushing down on the PCMCIA also doesn't cause any bad things to happen.
I assembled the A600, and I played several WHDload games for at least 30 minutes (previously I could hardly play for one minute without it crashing), after that I did 30 minutes of memtests without any errors, but shortly after that it froze as I opened a folder in WB. I'm sure the ACA620 was as hot as it would get at this point, so maybe it still needs some extra cooling?
Now it also feels a little better mechanically and there is less squeaking when installing it. I just recapped this machine and it had some acid damage in various places and I had to repair several traces and vias, so maybe some of the fumes from those caps also caused the CPU pins to oxidize, resulting in the bad connections that I had.
I will now try to sand off the CPU pins a bit with some grain 1000 to see if I can get the last 1% stability.